A hidden true state Θ is A or B, picked at random. Each agent, in turn, gets a private signal that matches Θ with probability p (the accuracy slider) — better than a coin flip, but never certain. Agents don't see each other's signals, only each other's public choices. A rational (Bayesian) agent weighs two pieces of evidence in log-odds terms: their own private signal, and everything the crowd's choices imply about Θ.
L_self = ±log(p / (1−p)) (own signal, sign = A or B)
L_crowd = (n_A − n_B) · log(p / (1−p)) (net public votes so far)
choose A if L_crowd + L_self > 0
choose B if L_crowd + L_self < 0
tie → fall back on own signal
Because every agent's signal carries the same weight log(p/(1−p)), two public votes in one direction always outweigh a single opposing private signal — for any p between 0.5 and 1. So the instant the running tally n_A − n_B reaches ±2, L_crowd alone decides the sign no matter what L_self says: the agent rationally stops using their own eyes and copies the majority. That's an information cascade — and once it starts, every later agent's action carries zero new private information, so the tally never resets. If the first couple of agents happened to draw wrong signals, the whole line can lock onto the wrong answer even though everyone behaved perfectly rationally.
- Private signal accuracy (p) — how reliable each individual's own evidence is. Higher p makes the correct signal more likely to "win" the crucial first few decisions, but the ±2-vote cascade threshold itself never changes.
- Sequence speed — how fast new agents join the queue; purely visual, doesn't change the outcome.
- Restart — redraws Θ and every private signal, so you can see how differently a cascade can lock in run after run.
This is the Bikhchandani–Hirshleifer–Welch herding model — it explains bank runs, fashion trends, and why the second and third restaurant reviewer often just repeats the first, all from purely rational, self-interested behaviour.